weyl group representations, nilpotent orbits, and …math.mit.edu/~dav/wolfho.pdfgelfand’s...

W -reps, nilp orbits,orbit method

David Vogan

Representationtheory

irr reps→ nilporbits

irr reps→W reps

nilp orbits↔Wreps

Explaining thearrows

Remembrance ofthings past

Weyl group representations, nilpotentorbits, and the orbit method

David Vogan

Department of MathematicsMassachusetts Institute of Technology

Lie groups: structure, actions and representationsIn honor of Joe Wolf, on his 75th birthday

January 11–14, 2012, Bochum


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Outline

What is representation theory about?

Nilpotent orbits from G reps

W reps from G reps

W reps and nilpotent orbits

What it all says about representation theory

The old good fan-fold days


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Gelfand’s “abstract harmonic analysis”

Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)

Method step 1: LINEARIZE. Replace M by Hilbertspace L2(M). Now G acts by unitary operators.Method step 2: DIAGONALIZE. Decompose L2(M)into minimal G-invariant subspaces.Method step 3: REPRESENTATION THEORY. Studyminimal pieces: irreducible unitary repns of G.What repn theory is about is 2 and 3.Today: what do irr unitary reps look like?


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Short version of the talk

Irr (unitary) rep of G! (coadjoint) orbit of G on g0∗.

!! fifty years of shattered dreams, broken promises

but I’m fine now, and not bitter

G reductive: coadjt orbit! conj class of matricesQuestions re matrices nilpotent matricesnilp matrices! combinatorics: partitions (or. . . )partitions (or. . . ) !Weyl grp reps

Conclusion:

irr unitary reps ←→ Weyl grp reps↘ ↗

nilp orbits

Plan of talk: explain the arrows.


David Vogan



irr reps→W reps

nilp orbits↔Wreps



When everything is easy ’cause of pCorresp G rep nilp coadjt orbit hard/R, easier/Qp.

G ⊂ GL(n,Qp) reductive alg, g0 ⊂ gl(n,Qp).

Put N ∗G = nilp elts of g∗0 , the nilpotent cone.

Orbit O has natural G-invt msre µO, homog deg dimO/2, hencetempered distribution; Fourier trans bµO = temp gen fn on g0.

Theorem (Howe, Harish-Chandra local char expansion)

π ∈ G, Θπ character (generalized function on G).

θπ = lift by exp to neighborhood of 0 ∈ g0.Then there are unique constants cO so that

θπ =∑O

cOµO πWF−→ {O|cO 6= 0}.

on some conj-invt nbhd of 0 ∈ g0.

Depends only on π restr to any small (compact) subgp.π restr to compact open

singularity of Θπ at e WF(π)


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Nilp orbits from G reps by analysis: WF(π)

(π,Hπ) irr rep HC−→ Θπ = char of π.Morally Θπ(g) = trπ(g); unitary op π(g) never traceclass, so get not function but “generalized function”:

Θπ(δ) = tr(∫

Gδ(g)π(g)

)(δ test density on G)

Singularity of Θπ (at origin) measuresinfinite-dimensionality of π.Howe definition:WF(π) = wavefront set of Θπ at e ⊂ T ∗

e (G) = g∗0passage Hπ →WF(π) is analytic classical limit.

WF(π) is G-invt closed cone of nilp elts in g∗0, so finiteunion of nilp coadjt orbit closures.


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Nilp orbits from G reps by comm alg: AV(π)

G real reductive Lie, K maximal compact subgp.

(π,Hπ) irr rep HC−→ HKπ Harish-Chandra module of

K -finite vectors; fin gen over U(g) (with rep of K ).Choose (arbitrary) fin diml gen subspace HK

π,0, define

HKπ,n =def Un(g) · HK

π,0, grHKπ =def

∞∑n=0

HKπ,n/HK

π,n−1.

grHKπ is fin gen over poly ring S(g) (with rep of K ).

AV(π) =def support of grHKπ

= K -invariant alg cone of nilp elts in (g/k)∗

⊂ g∗ = Spec S(g).

passage Hπ → AV(π) is algebraic classical limit.


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Interlude: real nilpotent cone

G real reductive Lie, K max compact, θ Cartan inv.g0 real Lie alg, g = g0 ⊗R C, G(C) cplx alg gp.K (C) = complexification of K : cplx reductive alg.N ∗ ⊂ g∗ = nilp cone; finite union of G(C) orbits.N ∗

R =def N ∗ ∩ g∗0; finite union of G orbits.N ∗θ =def N ∗ ∩ (g/k)∗; finite union of K (C) orbits.

Theorem (Kostant-Sekiguchi, Schmid-Vilonen).There’s natural bijection

N ∗R/G↔ N ∗

θ /K (C), WF(π)↔ AV(π).

Conclusion: two paths reps nilp coadj orbits,π 7→WF(π) and π 7→ AV(π) are same!


David Vogan



irr reps→W reps

nilp orbits↔Wreps



And now for something completely different. . .. . . to distinguish representations: Z(g) =def center of U(g).

π irr rep infinitesimal character ξ(π) : Z(g)→ C ishomomorphism giving action in π.

Fix Cartan subalgebra h ⊂ g, W = Weyl group.

HC: there’s bijection [infl chars ξλ : Z(g)]→ C↔ h∗/W .

Π(G)(λ) = (finite) set of irr reps of infl char ξλW (λ) = {w ∈W | wλ− λ = integer comb of roots}

Theorem (Lusztig-V). If λ is regular, then Π(G)(λ) hasnatural structure of W (λ)-graph. In particular,

1. W (λ) acts on free Z-module with basis Π(G(R))(λ).2. There’s preorder ≤LR on Π(G)(λ):

y ≤LR x ⇔ x appears in w · y (w ∈ W (λ))

⇔ π(x) subquo of π(y)⊗ F (F fin diml of Ad(G))

3. Each double cell (∼LR class) in Π(G)(λ) is W (λ)-graph, socarries W (λ)-repn.

4. WF(π(x)) constant for x in double cell.


David Vogan



irr reps→W reps

nilp orbits↔Wreps



W reps and nilpotent orbits

Nilp cone N ∗ = fin union of G(C) orbits O.

Springer corr W ↪→ {(O,S)}, (S loc sys on O).

Write σ 7→ (O(σ),S(σ)) (σ ∈ W ).Write a(σ) =lowest degree with σ ⊂ Sa(σ)(h)

1. a(σ) ≥ [dim(N ∗)− dim(O)]/2. Equality iff S(σ) trivial.2. For each O ∃! σ(O) ∈ W corr to (O, trivial).3. σ(O) has mult one in Sa(σ(O))(h).4. Every special rep of W (Lusztig) is of form σ(O).

W ⊃ Wnilpotent ⊃ Wspecial

Type An−1: all size p(n). E8: 112 cW ⊃ 70 cWnilp ⊃ 46 cWspecial.

σ ∈ W close to trivial⇔ a(σ) small⇔ O(σ) large.

σ ∈ W triv⇔ a(σ) = 0⇔ O(σ) = princ nilp.

σ ∈ W sgn⇔ a(σ) = |∆+| ⇔ O(σ) = zero nilp.


David Vogan



irr reps→W reps

nilp orbits↔Wreps



W (λ) reps and nilpotent orbitsλ ∈ h∗ ∆(λ) integral roots endoscopic gp G(λ)(C)

Weyl group W (λ), nilp cone N ∗(λ)).

σ(λ) ∈ W (λ)special ↔ O(λ) ⊂ N ∗(λ); codim = 2a(σ(λ)).

Proposition L ⊂ F fin gps, L 3 σL ⊂ X (reducible) rep of F .If σL has mult one in X then ∃! σ ∈ F , σL ⊂ σ ⊂ X .

W (λ) ⊂W , σ(λ) ⊂ Sa(σ(λ))(h) σ ∈ W , a(σ) = a(σ(λ)).

Theorem Representation σ ∈ W constructed from specialσ(λ) ∈ W (λ)special belongs to Wnilp. Get endoscopicinduction

special nilps for G(λ)(C) nilps for G(C),

preserves codimension in nilpotent cone.

O(λ) principal O principal

O(λ) = {0} O orbit for maxl prim ideal of infl char λ

G(λ)(C) Levi⇒ O(λ) O Lusztig-Spaltenstein induction


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Our story so far. . .

λ ∈ h∗ endoscopic gp G(λ)(C)

Block B of reps of G(R) of infl char λ

conn comp D of W (λ)-graph Π(G(R))(λ)

real form G(λ)(R); D ' D(λ) ⊂ Π(G(λ)(R))

Conclusion: for each double cell of reps of infl char λ

C ⊂ Π(G(R))(λ) σ ∈ cWnilp ↔ O ⊂ N ∗

l' ↘ ↑ ↑

C(λ) ⊂ Π(G(λ)(R)) ↔ σ(λ) ∈ W (λ)spec ↔ O(λ) ⊂ N ∗(λ)

Further conclusion: to understand G reps nilpotentorbs, must understand W graphs special W reps.

Real nilp orb(s) WF(C) (Howe) refine this correspondence.


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Special reps and W -graphsWant to understand W graphs special W reps.

Silently fix reg int infl char ξ (perhaps for endoscopic gp).x ∈W -graph = HC/Langlands param for irr of G(R)

= (H(R)x ,∆+x ,Λx ) mod G(R) conj

= (Hx ,∆+x , θx , (Z/2Z stuff)x ) mod G conj

= (Hp,∆+p , θx , (Z/2Z stuff)x )

Last step: move to (Cartan, pos roots) from pinning.

Z/2Z stuff grades θ-fixed roots as cpt/noncpt (real form)and −θ-fixed roots as nonparity/parity (block).

Rep of W on graph is sum (over real Cartans) of inducedfrom stabilizer of (θx ,Z/2Z stuff). . .

. . . so maps to sum of induced from stabilizer of θx .

W -graph for G(R)� quotient with basis {involutions}.Kottwitz calculated RHS ten years ago: for classical G it’ssum of all special reps of W , each with mult given byLusztig’s canonical quotient (of π1(O)).


David Vogan



irr reps→W reps

nilp orbits↔Wreps



What is to be done?

W -rep with basis {Langlands params}� quotientwith basis {involutions}.Kottwitz calculation of RHS explains (G(R) reps)→(complex special orbits).Lusztig-V (arxiv 2011): there’s a W -graph with vertexset {involutions}.PROBLEM: Relate LV W -graph structure oninvolutions to classical one on Langlands params.PROBLEM: refine Kottwitz calculation to includeZ/2Z stuff, to explain/calculate Howe’s wavefront setmap (G(R) reps)→ (real special orbits).


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Trying to outwit an old friend


David Vogan



irr reps→W reps

nilp orbits↔Wreps



Being outwitted by an old friend. . .


David Vogan



irr reps→W reps

nilp orbits↔Wreps



. . . who has old friends of his own. . .


David Vogan



irr reps→W reps

nilp orbits↔Wreps



. . . doesn’t know when to quit.

weyl group representations, nilpotent orbits, and …math.mit.edu/~dav/wolfho.pdfgelfand’s...

Documents